You are on an airplane traveling 30° south of due west at 130.0 m/s with respect to the air. The air is moving with a speed 31.0 m/s with respect to the ground due north.
1) What is the speed of the plane with respect to the ground?
2) What is the heading of the plane with respect to the ground? (Let 0° represent due north, 90° represents due easth).
3) How far west will the plane travel in 1 hour?
1) The speed of the plane with respect to the ground is 159.0 m/s.
2) The heading of the plane with respect to the ground is 60°.
3) The plane will travel 58,140 m west in 1 hour.
To solve these questions, we can break down the given information into components and then use vector addition to get the final answers.
Let's define the following:
- Velocity of the plane with respect to the air: V_air = 130.0 m/s 30° south of due west
- Velocity of the air with respect to the ground: V_ground = 31.0 m/s due north
1) To find the speed of the plane with respect to the ground, we need to find the resultant velocity of the plane by adding the velocities of the plane with respect to the air and the air with respect to the ground. We can use the concept of vector addition to do this.
To find the horizontal component (west-east direction) of the resultant velocity, we can use the cosine function:
V_x = V_air * cos(30°) (since the plane travels 30° south of due west)
To find the vertical component (north-south direction) of the resultant velocity, we can use the sine function:
V_y = V_air * sin(30°) (since the plane travels 30° south of due west)
Now, we can find the resultant velocity (V_resultant) using Pythagoras theorem:
V_resultant = sqrt(V_x^2 + V_y^2)
2) To find the heading of the plane with respect to the ground, we need to find the angle between the direction of the resultant velocity and due north.
We can use the tangent function to find this angle:
Angle = arctan(V_y / V_x) (make sure to convert the angle to the correct quadrant)
Note: The heading is measured from due north in a clockwise direction, so if the angle is negative, we need to add 360° to convert it to a positive value.
3) To find how far west the plane will travel in 1 hour, we can use the horizontal component of the resultant velocity.
Distance = V_x * time (with time = 1 hour)
Now, let's calculate the answers:
1) Speed of the plane with respect to the ground:
V_x = 130.0 m/s * cos(30°) = 112.4 m/s (approximately)
V_y = 130.0 m/s * sin(30°) = 65.0 m/s (approximately)
V_resultant = sqrt(V_x^2 + V_y^2) = sqrt((112.4 m/s)^2 + (65.0 m/s)^2) = 130.0 m/s (approximately)
Therefore, the speed of the plane with respect to the ground is 130.0 m/s.
2) Heading of the plane with respect to the ground:
Angle = arctan(V_y / V_x) = arctan(65.0 m/s / 112.4 m/s) = 30.0° (approximately)
Since the angle is positive, the heading of the plane with respect to the ground is 30° south of due west.
3) Distance traveled west in 1 hour:
Distance = V_x * time = 112.4 m/s * 1 hour = 112.4 km (approximately)
Therefore, the plane will travel approximately 112.4 km west in 1 hour.
To answer the questions, we will use vector addition.
1) To find the speed of the plane with respect to the ground, we need to calculate the magnitude of the resultant vector when we add the velocity of the plane with respect to the air to the velocity of the air with respect to the ground.
Using the Pythagorean theorem, we can calculate the magnitude of the resultant velocity:
v_plane_air = 130.0 m/s
v_air_ground = 31.0 m/s
v_ground = √(v_plane_air^2 + v_air_ground^2)
v_ground = √(130.0^2 + 31.0^2)
v_ground = √(16900 + 961)
v_ground = √17861
v_ground ≈ 133.7 m/s
Therefore, the speed of the plane with respect to the ground is approximately 133.7 m/s.
2) To find the heading of the plane with respect to the ground, we can use the tangent function:
tan(θ) = (v_air_ground / v_plane_air)
θ = arctan(v_air_ground / v_plane_air)
θ = arctan(31.0 / 130.0)
θ ≈ 13.9°
Therefore, the heading of the plane with respect to the ground is approximately 13.9° south of due west.
3) To find how far west the plane will travel in 1 hour, we can use the formula:
distance = velocity * time
Since the plane is traveling 30° south of due west, the horizontal component of its velocity is given by:
v_horizontal = v_ground * cos(θ)
v_horizontal = 133.7 * cos(30°)
v_horizontal ≈ 115.8 m/s
Therefore, the distance traveled westward in 1 hour will be:
distance = v_horizontal * time
distance = 115.8 m/s * 1 hour
distance ≈ 115.8 km
Therefore, the plane will travel approximately 115.8 kilometers westward in 1 hour.